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# Great retrosnub icosidodecahedron

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 Title: Great retrosnub icosidodecahedron Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Great retrosnub icosidodecahedron

Great retrosnub icosidodecahedron
Type Uniform star polyhedron
Elements F = 92, E = 150
V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5/2}
Wythoff symbol |3/2 5/3 2
Symmetry group I, [5,3]+, 532
Index references U74, C90, W117
Dual polyhedron Great pentagrammic hexecontahedron
Vertex figure
(34.5/2)/2
Bowers acronym Girsid

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It is given a Schläfli symbol s{3/2,5/3}.

## Cartesian coordinates

Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),
(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ

and

β = −ξ/τ+1/τ2−1/(ξτ),

where τ = (1+√5)/2 is the golden mean and ξ is the smaller positive real root of ξ3−2ξ=−1/τ, namely

\xi=\frac{\left(1+i \sqrt3\right)\left(\frac1{2 \tau}+\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13+ \left(1-i \sqrt3\right)\left(\frac1{2 \tau}-\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13}2

or approximately 0.3264046. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.